lecture 6 sharpening filters 1.the concept of sharpening filter 2.first and second order derivatives...
TRANSCRIPT
Lecture 6 Sharpening Filters
1. The concept of sharpening filter
2. First and second order derivatives
3. Laplace filter
4. Unsharp mask
5. High boost filter
6. Gradient mask
7. Sharpening image with MatLab
2
Sharpening Spatial Filters
• To highlight fine detail in an image or to enhance detail that has been blurred, either in error or as a natural effect of a particular method of image acquisition.
• Blurring vs. Sharpening
• Blurring/smooth is done in spatial domain by pixel averaging in a neighbors, it is a process of integration
• Sharpening is an inverse process, to find the difference by the neighborhood, done by spatial differentiation.spatial differentiation.
3
Derivative operator
• The strength of the response of a derivative operator is proportional to the degree of discontinuity of the image at the point at which the operator is applied.
• Image differentiation – enhances edges and other discontinuities (noise)– deemphasizes area with slowly varying gray-level
values.
4
Sharpening edge by First and second order derivatives
• Intensity function f =
• First derivative f ’ =
• Second-order derivative
f ’’ =
• f- f ’’ =
f
f’
f’’f-f’’
5
First and second order difference of 1D
• The basic definition of the first-order derivative of a one-dimensional function f(x) is the difference
• The second-order derivative of a one-dimensional function f(x) is the difference
)()1( xfxfx
f
)(2)1()1(2
2
xfxfxfx
f
7
First and Second-order derivative of 2D
• when we consider an image function of two variables, f(x, y), at which time we will dealing with partial derivatives along the two spatial axes.
y
yxf
x
yxf
yx
yxf
),(),(),(
f
2
2
2
22 ),(),(
y
yxf
x
yxff
(linear operator)
Laplacian operator(non-linear)
Gradient operator
8
Discrete form of Laplacian
),(2),1(),1(2
2
yxfyxfyxfx
f
),(2)1,()1,(2
2
yxfyxfyxfy
f
from
)],(4)1,()1,(
),1(),1([2
yxfyxfyxf
yxfyxff
9
Result Laplacian mask
10
Laplacian mask implemented an extension of diagonal neighbors
11
Other implementation of Laplacian masks
give the same result, but we have to keep in mind that when combining (add / subtract) a Laplacian-filtered image with another image.
12
Effect of Laplacian Operator• as it is a derivative operator,
– it highlights gray-level discontinuities in an image– it deemphasizes regions with slowly varying gray levels
• tends to produce images that have – grayish edge lines and other discontinuities, all superimposed on
a dark, – featureless background.
13
Correct the effect of featureless background
• easily by adding the original and Laplacian image.• be careful with the Laplacian filter used
),(),(
),(),(),(
2
2
yxfyxf
yxfyxfyxg
if the center coefficient of the Laplacian mask is negative
if the center coefficient of the Laplacian mask is positive
14
Example
• a). image of the North pole of the moon
• b). Laplacian-filtered image with
• c). Laplacian image scaled for display purposes
• d). image enhanced by addition with original image
1 1 1
1 -8 1
1 1 1
15
Mask of Laplacian + addition
• to simply the computation, we can create a mask which do both operations, Laplacian Filter and Addition the original image.
16
Mask of Laplacian + addition
)]1,()1,(
),1(),1([),(5
)],(4)1,()1,(
),1(),1([),(),(
yxfyxf
yxfyxfyxf
yxfyxfyxf
yxfyxfyxfyxg
0 -1 0
-1 5 -1
0 -1 0
17
Example
18
Note
0-1
0
-1
5-1
0-1
0
0 0 0
0 1 0
0 0 0
),(),(
),(),(),(
2
2
yxfyxf
yxfyxfyxg
= +0
-1
0
-1
4-1
0-1
0
0-1
0
-1
9-1
0-1
0
0 0 0
0 1 0
0 0 0= +
0-1
0
-1
8-1
0-1
0
19
Unsharp masking
• to subtract a blurred version of an image produces sharpening output image.
),(),(),( yxfyxfyxf s
sharpened image = original image – blurred imagesharpened image = original image – blurred image
Unsharp mask
22
High-boost filtering
• generalized form of Unsharp masking• A 1
),(),(),( yxfyxAfyxfhb
),(),()1(
),(),(),()1(),(
yxfyxfA
yxfyxfyxfAyxf
s
hb
23
High-boost filtering
• if we use Laplacian filter to create sharpen image fs(x,y) with addition of original image
),(),()1(),( yxfyxfAyxf shb
),(),(
),(),(),(
2
2
yxfyxf
yxfyxfyxf s
),(),(
),(),(),(
2
2
yxfyxAf
yxfyxAfyxfhb
24
High-boost Masks
A 1 if A = 1, it becomes “standard” Laplacian sharpening
25
Example
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Gradient Operator
• first derivatives are implemented using the magnitude of the gradientmagnitude of the gradient.
y
fx
f
G
G
y
xf
21
22
2122 ][)f(
y
f
x
f
GGmagf yx
the magnitude becomes nonlinearyx GGf
commonly approx.
27
Gradient Mask
• simplest approximation, 2x2z1 z2 z3
z4 z5 z6
z7 z8 z9
)( and )( 5658 zzGzzG yx
212
562
582
122 ])()[(][ zzzzGGf yx
5658 zzzzf
28
Gradient Mask
• Roberts cross-gradient operators, 2x2
z1 z2 z3
z4 z5 z6
z7 z8 z9
)( and )( 6859 zzGzzG yx
212
682
592
122 ])()[(][ zzzzGGf yx
6859 zzzzf
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Gradient Mask
• Sobel operators, 3x3
z1 z2 z3
z4 z5 z6
z7 z8 z9
)2()2(
)2()2(
741963
321987
zzzzzzG
zzzzzzG
y
x
yx GGf
the weight value 2 is to achieve smoothing by giving more important to the center point
30
Example
31
Example of Combining Spatial Enhancement Methods
• want to sharpen the original image and bring out more skeletal detail.
• problems: narrow dynamic range of gray level and high noise content makes the image difficult to enhance
32
Example of Combining Spatial Enhancement Methods
• solve :
1. Laplacian to highlight fine detail2. gradient to enhance prominent edges3. gray-level transformation to increase the dynamic
range of gray levels
